Painevg

2021-12-17

Find the points on the given curve where the tangent line is horizontal or vertical.

$r={e}^{\theta}$

encolatgehu

Beginner2021-12-18Added 27 answers

To find the points where the tangents are horizontal/vertical, we need to find dy/dx. Using the chain rule fore differentiation, we can write

Substitute

Use product rule for differentiation

Divide both the numerator and the denominator by

The tangents will be horizontal when the numerator is 0

Subtract

Divide both sides by

In the right-hand side, we will rewrite -1 as

Remember that: the general solution of the equation

Therefore, the tangents are horizontal when

The tangents will be vertical when the denomiantor is 0

Divides both sides by

In the right-hand side, we will rewrite 1 as

Therefore, the tangents are vertical when

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